Tipping and Stability
Find when a cart or mobile robot starts to tip.
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Support footprint
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Free-body diagram
Projection normal to the selected support edge
Selecting an edge updates both views and its moment balance.
Inspect force values & components
Force key: full 3D magnitudes. Select a label for its FBD components.
How to read these diagrams
The footprint looks straight onto the support plane, with front at the top and left on the left. G₀ is the mass-center projection, and R is the required ground-reaction point. Selecting a force highlights its application point in the footprint. The section shows height and slope at the selected edge. Its u axis always points into the footprint, so mirrored left/right cases can have the same section shape.
The calculation uses three-dimensional force vectors, application points, and moments. Ground contacts lie in one plane. The FBD projects those vectors onto a plane normal to the selected tipping edge, then rotates the whole section so projected weight points straight down. The apparent incline in this view can differ from the entered slope when the edge runs obliquely across the hill. G is the combined center of mass, W is weight, I is equivalent inertia, and P1, P2… are applied loads.
The surface-fixed axes are x forward, y left, and z normal to the ground. In the FBD, u points inward from the edge and z points away from the surface; both axes tilt with the ground. Geometry uses a linear scale. Arrow lengths are schematic, with force magnitudes listed in the key. Applied-force arrows start at their application points. Reaction arrows end at R, the required ground-reaction point. Only text labels move to avoid collisions; dotted leaders connect displaced labels.
Ground reactions are required resultants. Individual wheel load sharing and yaw capacity require a separate contact model.
Revisit the eight diagram studies or the JSXGraph interaction lab.
3D rigid-body model · tipping onset
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Explore results & calculations
The default cart is ready to calculate.
Support footprint
Surface coordinates · linearThe diamond is the required ground reaction. It reaches a support edge at the tipping threshold. The cross marks the center of mass projected normal to the surface.
Moments at every support edge
Select an edge to inspect its section and the moment from each load.
| Edge | Margin (m) | Reserve (N·m) |
|---|
| Load | Restoring moment (N·m) |
|---|
Gravity-only slope limits
Downhill direction is measured from forward toward left. Radius is critical slope in degrees, on a linear scale. Acceleration and applied forces are omitted from this plot.
1. The model
Ground contacts enclose a support polygon. The combined forces and moments determine where the ground's normal reaction must act. A point inside every edge has a positive tipping margin. At an edge, the restoring moment reserve is zero.
The model applies an equivalent inertial force opposite to the prescribed acceleration. This is a static or quasi-static equilibrium calculation. It assumes a fixed body configuration and sufficient restraint against rolling.
Suspension, pivoting axles, soft ground, noncoplanar contacts, curb impacts, angular inertia, and motion after tipping require additional models. A four-wheel footprint does not represent the stability triangle of a forklift with a pivoting axle.
2. Equations
3. Reading the results
Positive margin means the required normal reaction lies inside the modeled support polygon. It does not establish a rated operating limit. Sliding, rolling, local tire traction, ground bearing, and structural strength can limit the assembly separately.
The limit search starts with the selected parameter at zero and finds the first edge or contact boundary. Other loads stay fixed. A starting point outside the support polygon is reported explicitly; a later stabilizing load is not treated as a usable margin.
Individual normal forces at four or more contacts generally need a stiffness or load-sharing model. First individual wheel lift, tipping about an edge, and a completed overturn are different events.
4. Worked example
5. References
- Engineering Statics, §9.2: Slipping vs. tipping
- NHTSA DOT HS 811 486: Static Stability Factor
- OSHA: Stability of powered industrial trucks
These references explain the mechanics and their context. This tool does not implement an industry certification test. See the tool README for scope and verification.